
where \(M_1^{p,p+1}\)  and \(M_2^{p,p+1}\)  are local martingales. Again, since there are no indices \((k,\ell )\)  such that \(p\le k<\ell \le p+1,\ \ell -k>1\) , by (REF ) in Theorem REF  we have \(d\langle \lambda _i,\lambda _j\rangle _t=2t\delta _{ij}\) . Therefore, by Lévy's characterization for local martingales [1]}, there exist independent one-dimensional standard Brownian motions \(B_1^{p,p+1}(t),\ B_2^{p,p+1}(t)\)  such that \(M_1^{p,p+1}(t)=\sqrt{2}B_1^{p,p+1}(t),\ M_2^{p,p+1}(t)=\sqrt{2}B_2^{p,p+1}(t)\) , and we obtain (REF ). For the initial conditions, since
\(H_{\alpha _p}^{p,p+1}(0)=\begin{pmatrix}0 & x_p\\x_p & 0\end{pmatrix}\) , we have \(\lambda _1^{p,p+1}(0)=-x_p, \lambda _2^{p,p+1}(0)=x_p\) , and \(x_p>0\)  gives \(\lambda _1^{p,p+1}(0)<\lambda _2^{p,p+1}(0)\) , which actually satisfies (REF ).
